On the separation of simultaneous-source data by inversion |
There are many possible choices for the regularization operator . Taking to be an identity matrix and minimizing the regressions in equation 3 with a hybrid solver leads to a sparse Radon inversion problem. Alternatively, we can regularize the problem with a shot-space operator by re-writing equation 3 as follows:
In this paper, we define as a system of non-stationary dip-filters. First, we compute local event dips using the plane-wave destruction method (Fomel, 2002), then we compute dip-filters using factorized directional Laplacians (Hale, 2007). Because of the random delays between simultaneous sources, for any given source, events from other sources are random in its corresponding common-offset gathers. By destroying predictable events corresponding to source , operator ensures that only these events are allowed in the final separated data sets, whereas unpredictable events are not. Events that are not predictable by are passed on to other sources, where they are predictable by the corresponding operator . We call this inversion method dip-constrained sparse inversion (DCSI). In this paper, we refer to solution of equation 5, with as an identity matrix, as unconstrained sparse inversion.
However, because the operator is a function of the separated data, the problem becomes non-linear. To linearize this problem, we start by solving the equation 3 to get an initial estimate for . Then, using , we obtain an estimate of the operator , which is used to regularize the problem starting from initial model estimate . Results from this new step can then serve as inputs into the next inversion step. This process can be repeated as as many times as necessary.
Following the approach of Abma et al. (2010), instead of using as a regularization operator, we can use as a smoothing operator by re-writing equation 4 as follows:
Equation 5 can be directly extended to multiple surveys. For example, for two surveys, we can minimize the regressions
On the separation of simultaneous-source data by inversion |